Chapter 3 Geometry of convex functions - Meboo Publishing ...

236 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS
where the inverse always exists by (1427). This function is convex
simultaneously in both variables over the entire positive semidefinite cone S n +
and all x∈ R n : Consider Schur-form (1487) fromA.4: for t ∈ R
[ ] A + ǫI z
G(A, z , t) =
z T ǫ −1 ≽ 0
t
⇔
t − ǫz T (A + ǫI) −1 z ≥ 0
A + ǫI ≻ 0
(545)
Inverse image of the positive semidefinite cone S n+1
+ under affine mapping
G(A, z , t) is convex by Theorem 2.1.9.0.1. Function f(A, z) is convex on
S+×R n n because its epigraph is that inverse image:
epi f(A, z) = { (A, z , t) | A + ǫI ≻ 0, ǫz T (A + ǫI) −1 z ≤ t } = G −1( )
S n+1
+
(546)
3.6.1 matrix fractional projector function
Consider nonlinear function f having orthogonal projector W as argument:
f(W , x) = ǫx T (W + ǫI) −1 x (547)
Projection matrix W has property W † = W T = W ≽ 0 (1875). Any
orthogonal projector can be decomposed into an outer product of
orthonormal matrices W = UU T where U T U = I as explained in
E.3.2. From (1830) for any ǫ > 0 and idempotent symmetric W ,
ǫ(W + ǫI) −1 = I − (1 + ǫ) −1 W from which
Therefore
f(W , x) = ǫx T (W + ǫI) −1 x = x T( I − (1 + ǫ) −1 W ) x (548)
lim
ǫ→0 +f(W
, x) = lim (W + ǫI) −1 x = x T (I − W )x (549)
ǫ→0 +ǫxT
where I − W is also an orthogonal projector (E.2).
We learned from Example 3.6.0.0.4 that f(W , x)= ǫx T (W +ǫI) −1 x is
convex simultaneously in both variables over all x ∈ R n when W ∈ S n + is